Purpose In this activity, students explore multiplication and division using concrete and pictorial models in order to learn the meaning of these operations and how they are related.They will connect equal groups to arrays.
Prepare Connections Number Talk Scenarios (PG. 15) so that it can be projected using your classroom technology.
Make 1 copy of Connections Small Group Scenarios (PG. 16) for every 2–3 students.
Make 1 copy of Connections Independent Practice (PG. 17) for each student.
Write this journal question on the board: How are the scenarios we explored today the same? How are they different? Other materials:
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Scissors: 1 pair for every 2–3 students□
Glue sticks: 1 for every 2–3 students□
Markers and colored pencils: 1 pack for every 2–3 students□
Chart paper or butcher paper: 1 sheet for every 2–3 students□
Math journals
Teacher-facilitated w/ Small Student Groups Small GroupTutoring/Intervention Centers
Journal Anchor chartSetting Up For Instruction
Equal Groups
• Can you make a model/draw a picture to show your thinking? • How many are in each group?
• How many groups do you have? • What is the total number?
Arrays
• How many are in each row? • How many rows do you have? • How many are in each column? • How many columns do you have? • What is the product?
Connecting Equal Groups and Arrays
• How are equal groups and arrays alike?
EXPLORE MULTIPLICATION AND DIVISION CONTEXTS (WHOLE GROUP)
Goal: Represent multiplication and division contexts in numbers and pictures
1. Project Connections Number Talk Scenarios using your classroom technology.
2. Ask students to read the first scenario and think about how they would find the answer, including pictures and numbers. Record student thinking on chart paper.
How could you represent this situation using numbers or pictures? Add 5 four times; draw 4 groups of 5; skip count by 5s 4 times; draw 4 circles and put the number 5 in each one
3. Ask students to read the second scenario and think about how they would find the answer.
How could you represent this situation using numbers or pictures? Draw 20 lines, then circle them in groups of 5 to find 4 packs; subtract 5 from 20 until you reach zero, then see that you subtracted 5 four times, which means there are 4 packs 4. After students have discussed strategies for both problems, facilitate a discussion about how the scenarios are the same
or different.
How are these scenarios the same? How are they different? They both deal with equal numbers of gum; they can both be represented with pictures and numbers. In one scenario you know the total amount, in the other you are trying to find the total.
EXPLORE MULTIPLICATION AND DIVISION CONTEXTS (SMALL GROUP)
Goal: Represent multiplication and division contexts in numbers and pictures
5. Put students in groups of 2–3 and hand out chart paper and Connections Small Group Scenarios.
6. Have students cut Connections Small Group Scenarios in half. Have students glue the scenarios to the chart paper
and draw a T-Chart.
7. Have students work through both scenarios with their groups to represent each using pictures and numbers in as many ways as possible. Remind them to reference the chart paper from the opening whole-group activity for ideas.
8. Once groups are finished, ask each group to briefly share a strategy they used to represent and solve the problem. Facilitate a discussion about how the scenarios are the same or different.
How are these scenarios the same? How are they different? They both use the same numbers: 2, 9, 18 They both deal with equal groups of something; they can both be represented with pictures and numbers. In the first scenario you are trying to find the total, but in the second scenario you know the total and are trying to find the number of groups.
EXPLORE MULTIPLICATION AND DIVISION CONTEXTS (INDEPENDENT PRACTICE)
Goal: Match representations to multiplication and division scenarios 9. Hand out Connections Independent Practice to each student.
10. Review the directions. Model finding the first answer to Question 2 using Representation A and justifying your answer using words.
11. Have students match the remaining representations to each question and explain their reasoning.
12. Facilitate a discussion about how and why students selected which representations matched each question.
WRAP IT UP
Goal: Reflect on representing multiplication and division contexts
13. Ask students to reflect on the journal question in their math journals.
How-To Guide
There are three multiplication contexts students are likely to encounter in 3rd grade: finding the total, finding area, and
comparison. Helping students understand which models best fit each meaning of multiplication will help them efficiently select strategies to solve problems. Let’s take a look at each meaning and the models typically used in each situation. (More information on how to use each model can be found in the Content Extras in the next three activities.)
FINDING THE TOTAL
Situations where students are expected to multiply to find the total are the most common scenarios in 3rd grade. Finding the
Total multiplication problems reference objects in equal groups (oranges in baskets, cookies in bags) and ask students to find or represent how many total objects there are. Models that represent Finding the Total problems are repeated addition, equal-size groups, arrays, skip counting, equal jumps on a number line, and strip diagrams.
FINDING AREA
Area problems still involve finding a total number (square units), but with the understanding that these units are connected to each other with no overlap. Area models are the most accurate representation of these scenarios.
COMPARISON
Finally, comparison situations are specifically taught in 4th grade. Students must “interpret a multiplication equation as a comparison.” This means that an expression such as “3 × 24” means “3 times as much as 24.”
Students can represent comparison situations using several different types of models, but typically strip diagrams lend themselves to comparing two quantities side by side, so they often make the most accurate representation of these contexts. To be clear, flexibility in the use of multiple representations should be encouraged as students explore making meaning out of problem situations. Support students in making sense of the math in a way that makes sense to them. In upcoming activities, we’ll take a closer look at how to represent multiplication with each of these models so students have a wide variety of tools in their problem-solving toolbox from which to choose.
Different Meanings and Uses of Multiplication
Why are we putting multiplication and division together when 3rd graders are just learning to multiply and divide? As students solve problems using multiplication, they are also being informally introduced to how division could be used to represent or solve the same problems. The table below shows how multiplication and division facts are related.
Five groups of two turtles each equals ten total turtles. 5 × 2 = 10
Two groups of five turtles each equals ten total turtles. 2 × 5 = 10
Ten turtles divided into groups of two equals five total groups. 10 ÷ 2 = 5
Ten turtles divided into groups of five equals two total groups 10 ÷ 5 = 2
Since multiplication and division are inverse operations, we can use the same quantities (in this case, 2, 5, and 10) to create four related multiplication and division situations, also called “fact families.”
As students explore various representations of multiplication, many of them will make connections between the models they used for “equal sharing” in earlier grade levels and the multiplication models they’re using now. Encourage them to explore how they might connect models for multiplication to equal sharing scenarios, even if division isn’t being explicitly identified at that time. Then when the time comes to formally introduce division, students will already have a strong understanding of the concept, grounded in the work they have done with multiplication.
How do you define fluency in math? Many people think that fluency = speed; the faster a student is at solving problems or recalling facts, the more fluent they are. This is a widespread belief, but recent research indicates it’s having damaging effects on student performance and attitudes about math.
One of the most visible indicators of a “fluency = speed” mentality is timed fact tests. Researchers are finding that timed tests are a breeding ground for math anxiety. Math anxiety negatively impacts performance as students devote a portion of their working memory to attend to and manage the emotion system of their brain. This split attention limits their ability to process math, and can cause students to avoid engaging with math later in life to avoid their anxious feelings.
EXPERIMENT
There’s an easy way to feel what a math-anxious student feels during a timed test, even if you don’t have math anxiety yourself.
Follow the directions in each box.
Directions: Solve each problem. Set 1 37 × 14 = 59 × 36 = 68 × 42 = 85 × 53 = 71 × 20 = That wasn’t too hard, was it?
Directions: Memorize these letters.
AAQRNGFT
Directions: Cover the row of
letters above. Solve each problem. Then write the row of letters without looking. 73 × 41 = 95 × 63 = 86 × 24 = 58 × 35 = 17 × 15 = Write the letters.
Check your answers below.
Reflect: How was the experience of solving each problem set the same? How was it different? What impact did devoting part of your cognitive resources to remembering the letter strand have on your ability to complete the task? Did it impact your speed? What about your accuracy?
TIMED TESTS & MATH ANXIETY
On a small scale, this is what math-anxious students experience during timed tests. They have to give over precious working memory capacity to dealing with emotion and anxiety, which impedes their ability to focus on the task at hand.
Reducing or eliminating timed tests in your classroom is a great first step to improving student performance and reducing anxiety. If timed tests are required in your school or district, consider allowing students to write about their anxiety before testing to externalize it and diminish its impact. Here’s something else to try: discuss the symptoms of math anxiety with your students (e.g., rapid heart rate, etc.). Help them understand how to put their anxiety to work. Research shows that student performance often improves when we help them reframe anxiety (with its negative associations) as excitement (with its more positive associations). Who among us wouldn’t want to hear students saying, “I’m excited about math!”?
SHIFTING FROM TIMED TESTS TO NUMBER TALKS
A shift from timed tests to classroom routines like number talks allows students to hear each other’s thinking about math and encourages them to try to think flexibly, not fast, when problem solving. Routines like number talks also help to replace the “fluency = speed” illusion that can induce anxiety in students.
Ultimately, it’s about you, the teacher, and the message you communicate to students about what it means to be good at math. As you nurture students’ budding mathematical identities, be aware that just because timed tests are a long-standing practice does not mean that they are a best practice.
Fluency ≠ Speed Answ ers Set 1 : 518 ; 2,1 24; 2 ,856 ; 4,5 05; 1 ,42 0 Set 2 : 2,9 93; 5 ,98 5; 2 ,06 4; 2 ,03 0; 2 55
